2019年第11届全国大学生数学竞赛初赛(非数类)试题详细解答

分手后的思念是犯贱 2022-12-20 02:21 135阅读 0赞

>  *  [【2020年第12届全国大学生数学竞赛——资源分享 】【1~11届省赛、决赛考题及题解（数学类、非数学类）、推荐学习网址、复习备考书籍推荐】![icon-default.png?t=L892][icon-default.png_t_L892]https://blog.csdn.net/weixin\_44949135/article/details/109255209][2020_12_ _1_11_icon-default.png_t_L892_icon-default.png_t_L892_https_blog.csdn.net_weixin_44949135_article_details_109255209]
> 
> 1.  [视频地址：哔哩哔哩网站——2019年全国大学生数学竞赛（非数类）][2019]微信公众号：高数叔
> 2.  [视频地址：哔哩哔哩网站——第十一届全国大学生数学竞赛非数学类初赛真题讲解][Link 1]   微信公众号：考研数学竞赛
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70][]   ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 1][]

**目录**

P1 第一大题1（04:02）

解法一：比阶法

解法二：同一法

一站上浙——P1 11-1（09:36）

P2 第一大题2（04:04）

一站上浙——P2 11-2（09:28）

P3 第一大题3（03:34）

一站上浙——P3 11-3（12:54）

分部积分法 “反对幂指三”

三角函数 万能公式法

P4 第一大题4（07:06）

解法一

解法二

P5 第一大题5（04:48）

P6 第二大题（05:11）

P7 第三大题（11:42）

错误做法

解法一

解法二

解法三

P8 第四大题（07:39）

解法一

解法二

P9 第五大题（09:34）

P10 第六大题（09:35）

--------------------

# P1 第一大题1（04:02） #

## 解法一：比阶法 ##

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 2][]

## 解法二：同一法 ##

> [百度百科——对数公式][Link 2]   ![20201109202230388.png][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 3][]

## 一站上浙——P1 11-1（09:36） ##

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 4][]
> 
> ![20201122193635769.png][]
> 
> ![20201122193729307.png][]
> 
> ![20201122193748822.png][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 5][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 6][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 7][]

# P2 第一大题2（04:04） #

> 错误思路： ![20201109202817103.png][]   ![20201109202950269.png][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 8][]   ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 9][]
> 
> 注意：【y=tx也可以求得答案！】

## 一站上浙——P2 11-2（09:28） ##

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 10][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 11][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 12][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 13][]
> 
> ![20201122203700600.png][]![20201122203713445.png][]

# P3 第一大题3（03:34） #

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 14][]

## 一站上浙——P3 11-3（12:54） ##

### 分部积分法 “反对幂指三” ###

> ![20201122204359177.png][]![20201122204447307.png][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 15][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 16][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 17][]
> 
> 第1部分积分后的结果，更加复杂！
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 18][]

### 三角函数 万能公式法 ###

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 19][]
> 
> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 20][]

# P4 第一大题4（07:06） #

> ![20201109205016625.png][]

## 解法一 ##

> dx、dy：全微分方程                 简便计算：取(x0, y0)为(1, 0)。     第一种解法和格林公式以及高斯公式有关。
> 
> ![20201109210427940.png][]

## 解法二 ##

> 二次齐次式
> 
> ![2020110921103781.png][]

# P5 第一大题5（04:48） #

> ![20201109211111330.png][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 21][]

# P6 第二大题（05:11） #

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 22][]

> 球坐标      雅可比行列式      华莱士公式（点火公式）
> 
> ![20201110174328507.png][]

# P7 第三大题（11:42） #

> ![20201110174407898.png][]

## 错误做法 ##

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 23][]

## 解法一 ##

> ![20201110182310954.png][]

## 解法二 ##

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 24][]

## 解法三 ##

> ![20201110182900994.png][]

# P8 第四大题（07:39） #

> ![20201110182952478.png][]

## 解法一 ##

> 雅可比行列式
> 
> ![20201110184340952.png][]

## 解法二 ##

> ![20201110184308490.png][]

# P9 第五大题（09:34） #

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 25][]

>  ![20201110190341819.png][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 26][]

# P10 第六大题（09:35） #

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 27][]

> ![watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NDk0OTEzNQ_size_16_color_FFFFFF_t_70 28][]

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[2020_12_ _1_11_icon-default.png_t_L892_icon-default.png_t_L892_https_blog.csdn.net_weixin_44949135_article_details_109255209]: /images/20221120/82707dac12794dc3a26af54d78ab4c49.png
[2019]: https://www.bilibili.com/video/BV16V411o7sB
[Link 1]: https://www.bilibili.com/video/BV1Qa4y1s77t
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[Link 2]: https://baike.baidu.com/item/%E5%AF%B9%E6%95%B0%E5%85%AC%E5%BC%8F/5557846?fr=aladdin
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